# Hubbard model diagonalization in 1D K-space for spinless Fermions

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• Luqman Saleem
In summary: Another helpful resource is “The Hubbard Model: A Simple Model of Strongly Correlated Electrons” by F. F. Assaad, published in the European Physical Journal B in 2003.
Luqman Saleem
I am trying to diagonalize hubbard model in real and K-space for spinless fermions. Hubbard model in real space is given as:
$$H=-t\sum_{<i,j>}(c_i^\dagger c_j+h.c.)+U\sum (n_i n_j)$$
I solved this Hamiltonian using MATLAB. It was quite simple. t and U are hopping and interaction potentials. c, $$c^\dagger$$ and n are annihilation, creation and number operators in real space respectively. The first term is hopping and 2nd is two-body interaction term. <i,j> is indicating that hopping is possible only to nearest neighbors. To solve this Hamiltonian I break it down as: (for M=# for sites=2 and N=# of particles=1)
$$H=-t (c_1^\dagger c_2 + c_2^\dagger c_1)+U n_1 n_2$$
The basis vectors that can be written in binary notation are:
01, 10
Using t=1, U=1 and above basis the Hamiltonian can be written as:
H=[0 -1
-1 0]
That is correct.
I checked with different values of M,N,U and t this MATLAB program give correct results.

$$**In K-space**$$
To diagonalize this Hamiltonian in K-space we can perform Fourier transform of operators that will results in:
$$H(k)=\sum_k \epsilon_k n_k + U / L \sum_ {k,k,q} c_k^\dagger c_{k-q} c_{k'}^\dagger c_{k'+q}$$
Where $$\epsilon_k=-2tcos(k)$$.
To diagonalize this Hamiltonian I make basis by taking k-points between -pi and +pi (first brillion zone) i.e. for M=2 and N=1 allowed k-points are: [0,pi]
Here first term is simple to solve and I have solved it already but I can't solve the 2nd term as it includes summation over three variables.
To get in more details of my attempt you can see https://physics.stackexchange.com/q/352833/140141

$$**My question:**$$
1. What is physical significance of 2nd term in H(k) given above? I mean what is it telling about which particles are hopping from where to where? What are limits on q, k and k'?
2. If you think any article can help me with this problem then please tell me about that.Thanks a lot.

1. The second term in the Hamiltonian describes the scattering between two particles with momentum k and k' due to an interaction U. The limit on q is that it must be less than the first Brillouin zone (i.e. q must be between -pi and +pi). The limits on k and k' are determined by the wave vector of the two particles, i.e. they must both be within the allowed range of wave vectors.2. There are several articles which discuss the diagonalization of the Hubbard model in k-space. One example is “Diagonalization of the Hubbard Model in K Space” by A. G. Green, published in the Journal of Mathematical Physics in 1998.

## 1. What is the Hubbard model diagonalization in 1D K-space for spinless Fermions?

The Hubbard model is a theoretical model used to describe the behavior of interacting particles in a lattice. In 1D K-space, the model is used to study the motion of spinless fermions (particles with half-integer spin) in a one-dimensional lattice structure. Diagonalization refers to the process of finding the eigenvalues and eigenvectors of the Hamiltonian matrix for this system, which allows for the calculation of various physical properties.

## 2. Why is the Hubbard model used in 1D K-space for spinless Fermions?

The Hubbard model is often used to study the behavior of particles in a lattice because it takes into account the effects of particle interactions, which can have a significant impact on the system's properties. In 1D K-space, the model is particularly useful for understanding the behavior of spinless fermions, as it allows for the study of their motion and correlations in this simplified system.

## 3. How is the diagonalization process carried out in 1D K-space for spinless Fermions?

The diagonalization of the Hubbard model in 1D K-space is typically carried out using numerical methods, such as exact diagonalization or density matrix renormalization group (DMRG) techniques. These methods involve constructing the Hamiltonian matrix for the system and then finding its eigenvalues and eigenvectors using computer algorithms.

## 4. What are some physical properties that can be calculated using the Hubbard model diagonalization in 1D K-space for spinless Fermions?

The diagonalization of the Hubbard model in 1D K-space allows for the calculation of various physical properties, such as the ground state energy, the correlation functions, and the excitation spectrum of the system. These properties can provide insight into the behavior of spinless fermions in a one-dimensional lattice and how they interact with each other.

## 5. What are some potential applications of the results obtained from the Hubbard model diagonalization in 1D K-space for spinless Fermions?

The results obtained from the diagonalization of the Hubbard model in 1D K-space can have applications in various fields, such as condensed matter physics, materials science, and quantum information. They can help researchers understand the behavior of electrons in solids and potentially lead to the development of new materials with specific properties. In quantum information, the diagonalization results can be used to study the entanglement and quantum correlations of spinless fermions in a lattice structure.

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